uh... hmmm... well...
yes, a noise gate is the right plug-in to try to remove any not-as-loud-as-the-wanted-signal in-between noises.
The typical use case for a noise gate are multiple single microphones (think of a drumset with a mic on each drum).
You would use a noise gate to make sure the channels go quiet except for a hit on the particular drum - without a noise gate, you would have that one drum coming thru all of your mics, making it harder to mix (such channel crossing noises are called "bleeding" btw).
That being said, a noise gate is quite a tricky thing to set up properly, because you want to make sure it "opens" as soon as you start to play, even if you don't play at full throttle.
Your described pick noise will probably not be easy to gate,
you may find out a gate set up to remove the pick noise will also remove moments when you play softly ...
you may try, but I don't think it'll work well.

I'd rather look at that pick noise as something which you should try to control in your playing- these noises are a totally important part of your individual sound (like the breathing of a vocalist: remove the breathing, and half of the performances magic is gone!).
If you can get control over the "noises" in your playing without needing a plug-in, then that will improve your recordings value much more than a noise gate ever will IMHO.
Hope this makes sense & happy experiments :)

If you want to remove clicks made from the pick at the start of every 'attack' you make you might want to look at a plug-in called Freiraum. There's an option called entropy which you can alter to emphasize either the pick noise or the tone that's generated.

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